A converse to the Andreotti-Grauert theorem

Jean-Pierre Demailly[1]

  • [1] Université de Grenoble I, Département de Mathématiques, Institut Fourier, 38402 Saint-Martin d’Hères, France

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: S2, page 123-135
  • ISSN: 0240-2963

Abstract

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The goal of this paper is to show that there are strong relations between certain Monge-Ampère integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of holomorphic line bundles. Especially, we prove that these relations hold without restriction for projective surfaces, and in the special case of the volume, i.e. of asymptotic 0 -cohomology, for all projective manifolds. These results can be seen as a partial converse to the Andreotti-Grauert vanishing theorem.

How to cite

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Demailly, Jean-Pierre. "A converse to the Andreotti-Grauert theorem." Annales de la faculté des sciences de Toulouse Mathématiques 20.S2 (2011): 123-135. <http://eudml.org/doc/219730>.

@article{Demailly2011,
abstract = {The goal of this paper is to show that there are strong relations between certain Monge-Ampère integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of holomorphic line bundles. Especially, we prove that these relations hold without restriction for projective surfaces, and in the special case of the volume, i.e. of asymptotic $0$-cohomology, for all projective manifolds. These results can be seen as a partial converse to the Andreotti-Grauert vanishing theorem.},
affiliation = {Université de Grenoble I, Département de Mathématiques, Institut Fourier, 38402 Saint-Martin d’Hères, France},
author = {Demailly, Jean-Pierre},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {asymptotic cohomology functions; holomorphic Morse inequalities; volume of a line bundle},
language = {eng},
month = {4},
number = {S2},
pages = {123-135},
publisher = {Université Paul Sabatier, Toulouse},
title = {A converse to the Andreotti-Grauert theorem},
url = {http://eudml.org/doc/219730},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Demailly, Jean-Pierre
TI - A converse to the Andreotti-Grauert theorem
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/4//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - S2
SP - 123
EP - 135
AB - The goal of this paper is to show that there are strong relations between certain Monge-Ampère integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of holomorphic line bundles. Especially, we prove that these relations hold without restriction for projective surfaces, and in the special case of the volume, i.e. of asymptotic $0$-cohomology, for all projective manifolds. These results can be seen as a partial converse to the Andreotti-Grauert vanishing theorem.
LA - eng
KW - asymptotic cohomology functions; holomorphic Morse inequalities; volume of a line bundle
UR - http://eudml.org/doc/219730
ER -

References

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  1. Andreotti (A.), Grauert (H.).— Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90, p. 193-259 (1962). Zbl0106.05501MR150342
  2. Angelini (F.).— An algebraic version of Demailly’s asymptotic Morse inequalities; arXiv: alg-geom/9503005, Proc. Amer. Math. Soc. 124 p. 3265-3269 (1996). Zbl0860.14019MR1389502
  3. Boucksom (S.).— On the volume of a line bundle, Internat. J. Math. 13, p. 1043-1063 (2002). Zbl1101.14008MR1945706
  4. Boucksom (S.), Demailly (J.-P.), Păun (M.), Peternell (Th.).— The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, arXiv: math.AG/0405285, see also Proceedings of the ICM 2006 in Madrid. MR1351504
  5. Demailly (J.-P.).— Estimations L 2 pour l’opérateur ∂ ¯ d’un fibré vectoriel holomorphe semi-positif au dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. 15, p. 457-511 (1982). Zbl0507.32021MR690650
  6. Demailly (J.-P.).— Champs magnétiques et inégalités de Morse pour la d ′ ′ -cohomologie, Ann. Inst. Fourier (Grenoble) 35, p. 189-229 (1985). Zbl0565.58017MR812325
  7. Demailly (J.-P.).— Holomorphic Morse inequalities, Lectures given at the AMS Summer Institute on Complex Analysis held in Santa Cruz, July 1989, Proceedings of Symposia in Pure Mathematics, Vol. 52, Part 2, p. 93-114 (1991). Zbl0755.32008MR1128538
  8. Demailly (J.-P.).— Regularization of closed positive currents and Intersection Theory, J. Alg. Geom. 1, p. 361-409 (1992). Zbl0777.32016MR1158622
  9. Demailly (J.-P.).— Holomorphic Morse inequalities and asymptotic cohomology groups: a tribute to Bernhard Riemann, Milan Journal of Mathematics 78, p. 265-277 (2010). Zbl1205.32017MR2684780
  10. Demailly (J.-P.), Ein (L.) and Lazarsfeld (R.).— A subadditivity property of multiplier ideals, Michigan Math. J. 48, p. 137-156 (2000). Zbl1077.14516MR1786484
  11. Demailly (J.-P.), Păun (M.).— Numerical characterization of the Kähler cone of a compact Kähler manifold, arXiv: math.AG/0105176 ; Annals of Math. 159, p. 1247-1274 (2004). Zbl1064.32019MR2113021
  12. de Fernex (T.), Küronya (A.), Lazarsfeld (R.).— Higher cohomology of divisors on a projective variety, Math. Ann. 337, p. 443-455 (2007). Zbl1127.14012MR2262793
  13. Fujita (T.).— Approximating Zariski decomposition of big line bundles, Kodai Math. J. 17, p. 1-3 (1994). Zbl0814.14006MR1262949
  14. Hironaka (H.).— Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79, p. 109-326 (1964). Zbl0122.38603MR199184
  15. Küronya (A.).— Asymptotic cohomological functions on projective varieties, Amer. J. Math. 128, p. 1475-1519 (2006). Zbl1114.14005MR2275909
  16. Lazarsfeld (R.).— Positivity in Algebraic Geometry I.-II, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vols. 48-49, Springer Verlag, Berlin, 2004. Zbl1093.14500MR2095471
  17. Totaro (B.).— Line bundles with partially vanishing cohomology, July 2010, arXiv: math.AG/1007.3955. 

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